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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist"></a><a class="link" href="inverse_gaussian_dist.html" title="Inverse Gaussian (or Inverse Normal) Distribution">Inverse
        Gaussian (or Inverse Normal) Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">inverse_gaussian</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">inverse_gaussian_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
   <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>

   <span class="identifier">inverse_gaussian_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">mean</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>

   <span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// mean default 1.</span>
   <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Optional scale, default 1 (unscaled).</span>
   <span class="identifier">RealType</span> <span class="identifier">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="comment">// Shape = scale/mean.</span>
<span class="special">};</span>
<span class="keyword">typedef</span> <span class="identifier">inverse_gaussian_distribution</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">inverse_gaussian</span><span class="special">;</span>

<span class="special">}}</span> <span class="comment">// namespace boost // namespace math</span>
</pre>
<p>
          The Inverse Gaussian distribution distribution is a continuous probability
          distribution.
        </p>
<p>
          The distribution is also called 'normal-inverse Gaussian distribution',
          and 'normal Inverse' distribution.
        </p>
<p>
          It is also convenient to provide unity as default for both mean and scale.
          This is the Standard form for all distributions. The Inverse Gaussian distribution
          was first studied in relation to Brownian motion. In 1956 M.C.K. Tweedie
          used the name Inverse Gaussian because there is an inverse relationship
          between the time to cover a unit distance and distance covered in unit
          time. The inverse Gaussian is one of family of distributions that have
          been called the <a href="http://en.wikipedia.org/wiki/Tweedie_distributions" target="_top">Tweedie
          distributions</a>.
        </p>
<p>
          (So <span class="emphasis"><em>inverse</em></span> in the name may mislead: it does <span class="bold"><strong>not</strong></span> relate to the inverse of a distribution).
        </p>
<p>
          The tails of the distribution decrease more slowly than the normal distribution.
          It is therefore suitable to model phenomena where numerically large values
          are more probable than is the case for the normal distribution. For stock
          market returns and prices, a key characteristic is that it models that
          extremely large variations from typical (crashes) can occur even when almost
          all (normal) variations are small.
        </p>
<p>
          Examples are returns from financial assets and turbulent wind speeds.
        </p>
<p>
          The normal-inverse Gaussian distributions form a subclass of the generalised
          hyperbolic distributions.
        </p>
<p>
          See <a href="http://en.wikipedia.org/wiki/Normal-inverse_Gaussian_distribution" target="_top">distribution</a>.
          <a href="http://mathworld.wolfram.com/InverseGaussianDistribution.html" target="_top">Weisstein,
          Eric W. "Inverse Gaussian Distribution." From MathWorld--A Wolfram
          Web Resource.</a>
        </p>
<p>
          If you want a <code class="computeroutput"><span class="keyword">double</span></code> precision
          inverse_gaussian distribution you can use
        </p>
<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_gaussian_distribution</span><span class="special">&lt;&gt;</span></pre>
<p>
          or, more conveniently, you can write
        </p>
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">inverse_gaussian</span><span class="special">;</span>
<span class="identifier">inverse_gaussian</span> <span class="identifier">my_ig</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span>
</pre>
<p>
          For mean parameters μ and scale (also called precision) parameter λ, and random
          variate x, the inverse_gaussian distribution is defined by the probability
          density function (PDF):
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">f(x;μ, λ) = √(λ/2πx<sup>3</sup>) e<sup>-λ(x-μ)²/2μ²x</sup> </span>
          </p></blockquote></div>
<p>
          and Cumulative Density Function (CDF):
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">F(x;μ, λ) = Φ{√(λ<span class="emphasis"><em>x) (x</em></span>μ-1)}
            + e<sup>2μ/λ</sup> Φ{-√(λ/μ) (1+x/μ)} </span>
          </p></blockquote></div>
<p>
          where Φ is the standard normal distribution CDF.
        </p>
<p>
          The following graphs illustrate how the PDF and CDF of the inverse_gaussian
          distribution varies for a few values of parameters μ and λ:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_pdf.svg" align="middle"></span>

          </p></blockquote></div>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/inverse_gaussian_cdf.svg" align="middle"></span>

          </p></blockquote></div>
<p>
          Tweedie also provided 3 other parameterisations where (μ and λ) are replaced
          by their ratio φ = λ/μ and by 1/μ: these forms may be more suitable for Bayesian
          applications. These can be found on Seshadri, page 2 and are also discussed
          by Chhikara and Folks on page 105. Another related parameterisation, the
          __wald_distrib (where mean μ is unity) is also provided.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.member_functions"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">inverse_gaussian_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">df</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// optionally scaled.</span>
</pre>
<p>
          Constructs an inverse_gaussian distribution with μ mean, and scale λ, with
          both default values 1.
        </p>
<p>
          Requires that both the mean μ parameter and scale λ are greater than zero,
          otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">mean</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the mean μ parameter of this distribution.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the scale λ parameter of this distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.non_member_accessors"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.non_member_accessors">Non-member
          Accessors</a>
        </h5>
<p>
          All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
          functions</a> that are generic to all distributions are supported:
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
        </p>
<p>
          The domain of the random variate is [0,+∞).
        </p>
<div class="note"><table border="0" summary="Note">
<tr>
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../doc/src/images/note.png"></td>
<th align="left">Note</th>
</tr>
<tr><td align="left" valign="top"><p>
            Unlike some definitions, this implementation supports a random variate
            equal to zero as a special case, returning zero for both pdf and cdf.
          </p></td></tr>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.accuracy"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The inverse_gaussian distribution is implemented in terms of the exponential
          function and standard normal distribution <span class="emphasis"><em>N</em></span>0,1 Φ : refer
          to the accuracy data for those functions for more information. But in general,
          gamma (and thus inverse gamma) results are often accurate to a few epsilon,
          &gt;14 decimal digits accuracy for 64-bit double.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.implementation"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table μ is the mean parameter and λ is the scale parameter
          of the inverse_gaussian distribution, <span class="emphasis"><em>x</em></span> is the random
          variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>
          its complement. Parameters μ for shape and λ for scale are used for the inverse
          gaussian function.
        </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Function
                  </p>
                </th>
<th>
                  <p>
                    Implementation Notes
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    pdf
                  </p>
                </td>
<td>
                  <p>
                    √(λ/ 2πx<sup>3</sup>) e<sup>-λ(x - μ)²/ 2μ²x</sup>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Φ{√(λ<span class="emphasis"><em>x) (x</em></span>μ-1)} + e<sup>2μ/λ</sup> Φ{-√(λ/μ) (1+x/μ)}
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    using complement of Φ above.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    No closed form known. Estimated using a guess refined by Newton-Raphson
                    iteration.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    No closed form known. Estimated using a guess refined by Newton-Raphson
                    iteration.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    μ {√(1+9μ²/4λ²)² - 3μ/2λ}
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    median
                  </p>
                </td>
<td>
                  <p>
                    No closed form analytic equation is known, but is evaluated as
                    quantile(0.5)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    μ
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    variance
                  </p>
                </td>
<td>
                  <p>
                    μ³/λ
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    3 √ (μ/λ)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis_excess
                  </p>
                </td>
<td>
                  <p>
                    15μ/λ
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    12μ/λ
                  </p>
                </td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.h4"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.inverse_gaussian_dist.references"></a></span><a class="link" href="inverse_gaussian_dist.html#math_toolkit.dist_ref.dists.inverse_gaussian_dist.references">References</a>
        </h5>
<div class="orderedlist"><ol class="orderedlist" type="1">
<li class="listitem">
              Wald, A. (1947). Sequential analysis. Wiley, NY.
            </li>
<li class="listitem">
              The Inverse Gaussian distribution : theory, methodology, and applications,
              Raj S. Chhikara, J. Leroy Folks. ISBN 0824779975 (1989).
            </li>
<li class="listitem">
              The Inverse Gaussian distribution : statistical theory and applications,
              Seshadri, V , ISBN - 0387986189 (pbk) (Dewey 519.2) (1998).
            </li>
<li class="listitem">
              <a href="http://docs.scipy.org/doc/numpy/reference/generated/numpy.random.wald.html" target="_top">Numpy
              and Scipy Documentation</a>.
            </li>
<li class="listitem">
              <a href="http://bm2.genes.nig.ac.jp/RGM2/R_current/library/statmod/man/invgauss.html" target="_top">R
              statmod invgauss functions</a>.
            </li>
<li class="listitem">
              <a href="http://cran.r-project.org/web/packages/SuppDists/index.html" target="_top">R
              SuppDists invGauss functions</a>. (Note that these R implementations
              names differ in case).
            </li>
<li class="listitem">
              <a href="http://www.statsci.org/s/invgauss.html" target="_top">StatSci.org invgauss
              help</a>.
            </li>
<li class="listitem">
              <a href="http://www.statsci.org/s/invgauss.statSci.org" target="_top">invgauss
              R source</a>.
            </li>
<li class="listitem">
              <a href="http://www.biostat.wustl.edu/archives/html/s-news/2001-12/msg00144.html" target="_top">pwald,
              qwald</a>.
            </li>
<li class="listitem">
              <a href="http://www.brighton-webs.co.uk/distributions/wald.asp" target="_top">Brighton
              Webs wald</a>.
            </li>
</ol></div>
</div>
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      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
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